Vlasov moment flows and geodesics on the Jacobi group (Q2872447)

The author used the polynomial algebra \(\mathrm{Pol}(\mathbb{R}^{2k}) \subset \mathcal{F}(\mathbb{R}^{2k})\) associated to the Vlasov equation in order to identify the Jacobi group as the Lie symmetry underlying the integrable Bloch-Iserles system. It is shown that the general case when \(\operatorname{corank}(\mathbb{N}) > 1\) requires a suitable generalization of the Jacobi group that is different from other generalizations already available in the literature. The Lie-Poisson structure of the Bloch-Iserles system is then analyzed in detail and it is shown how this structure leads to a natural momentum map, which in turn arises from the Vlasov single-particle solution. The relation between Vlasov dynamics and the Bloch-Iserles system naturally leads to the question of how the Jacobi group is a subgroup of the strict contact transformations, which underlie Vlasov dynamics. This question is addressed in the second part of this paper, where explicit subgroup inclusions are presented. Also, the authors show how the generalized Jacobi group is a natural generalization of \(\mathrm{Cont}(\mathbb{R}^{2k+1}, \theta)\). After the relation between the Bloch-Iserles system and its underlying Vlasov description is completely characterized, the momentum maps accompanying the Klimontovich solution are analyzed in detail.